Scrambling of locally perturbed thermal states Joan Sim on - - PowerPoint PPT Presentation

Scrambling of locally perturbed thermal states

Joan Sim´

• n

University of Edinburgh and Maxwell Institute of Mathematical Sciences

Topics in 3D Gravity ICTP, Trieste, March 22nd 2016 Based on arXiv:1503.08161 with P. Caputa, A. ˇ Stikonas,

• T. Takayanagi & K. Watanabe

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Outline

1

Motivating & quantifying scrambling

2

2d CFT discussion

◮ Set-up ◮ Large c 2d CFTs at finite T & thermo field double 3

Brief holographic discussion

4

Quantum chaos & butterfly effect vs scrambling

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Motivation

Question If we input some localised information in a quantum system, such as a perturbation, does it remain localised or does it spread over the entire system ? ∃ delocalisation ∼ scrambling Measures of scrambling

1

Any arbitrary subsystem up to half of the state’s dof is nearly maximally entangled : Page scrambling

2

If ∃ scrambling ⇒ information about input can not be deduced by local output measurements ⇒ mutual information may quantify this

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Quantifying scrambling

Any unitary operator U(t) =

2n−1

• i,j=0

uij|ij| can be mapped into a 2n-qubit state treating input and output legs equally |U(t) = 1 2n/2

2n−1

• i,j=0

uij|iin ⊗ |iout This is an example of the channel-state duality in QI.

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Quantifying scrambling

Given some local disturbance in A, ∃ scrambling ⇒ not measurable in C ⇓ I(A : C) = SA+SC−SA∪C small

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Quantifying scrambling

Same argument and conclusion for local region D ⇒ I(A : D) small Amount of information that is non-locally hidden in CD by computing I(A : CD) − I(A : C) − I(A : D) In QI, this is captured by the tripartite information I3(A : C : D) = I(A : C) + I(A : D) − I(A : CD) being very negative (Hosur, Qi, Roberts & Yoshida) Today, we will focus on I(A : C) ≈ 0

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BHs : scrambling vs quantum cloning

BH physics suggest speed at which thermality is regained is faster than in diffusive systems (scrambling) (Susskind-Sekino)

1

scrambling time from causality bound preventing quantum cloning τscrambling ∼ β log S

2

Faster than diffusion τdiff ∼ S2/d ≫ log S

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Perturbing eternal BH (Shenker & Stanford)

Perturbation turned on at time t1 on the left boundary Backreaction can be non-trivial, no matter how light the perturbation is, depending on the t1 scale Shock-wave description

M M M+E M+E M+E t1 t1

α

Small perturbations get blue shifted near horizon (Shenker-Stanford) t⋆ ∼ β log mp β

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CFT calculation

Consider a 2d CFT at finite temperature ρβ Perturb the thermal state by a local primary operator Ow(x0, 0) ρβ O†

w(x0, 0)

Evolve the system unitarily e−iHt Ow(x0, 0) ρβ O†

w(x0, 0)

• eiHt

Question Time scale t⋆ at which I(A : B)(t⋆) = 0

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CFT : overall strategy

Non-compact 2d CFT ⇒ no Poincar´ e recurrences Logic :

1

Describe density operator and its regularisation (on top of the UV cut-off)

2

Use replica trick to compute entanglement entropy ⇒ correlators in 2d CFT

3

Use large c limit to compute these correlators analytically

4

Solve for the scrambling time scale t⋆ Remark : this main set-up was also considered in Roberts & Stanford

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CFT set-up

Consider a perturbation, generated at time t = 0 at x = −ℓ, created by a primary operator O acting on the vacuum of the 2d CFT : |ΨO(t) = √ N e−iHt e−ǫHO(0, −ℓ)|0 O is inserted at t = 0 and x = −ℓ and dynamically evolved afterwards ǫ is a small parameter smearing the UV behaviour of the local

• perator (separation in euclidean time)

Density matrix : ρ(t) = N e−iHt e−ǫHO(0, −ℓ)|00|O†(0, −ℓ) eiHt e−ǫH = N O(ω2, ¯ ω2)|00|O†(ω1, ¯ ω1) where ω1 = −ℓ + i(ǫ − it), ω2 = −ℓ − i(ǫ + it) (¯ ω1 = −ℓ − i(ǫ − it))

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Entanglement entropy : replica trick

Method 1 : uniformization Trρn

A ∼ O(ω1, ¯

ω1)O†(ω2, ¯ ω2)...O†(ω2n, ¯ ω2n)Σn

◮ Σn Riemann surface ◮ ω2k+1 = e2πikω1 ◮ ω2k+2 = e2πikω2

Method 2 : Twist operators Trρn

A ∼ ψ|σ(ω1, ¯

ω1)˜ σ(ω2, ¯ ω2)|ψ

◮ Calculation done in n-copies of the original CFT ◮ Twist operators emerge because of the existence of some internal

symmetry when swapping these copies

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Perturbations at finite temperature

Same set-up as before, but now

1

we perturb a thermal state at t = −tω : ρ(t) ≡ NO(ω2, ¯ ω2) e−βH O†(ω1, ¯ ω1) with ω1 = x0 + t + tω + iǫ ¯ ω1 = x0 − t − tω − iǫ ω2 = x0 + t + tω − iǫ ¯ ω2 = x0 − t − tω + iǫ .

2

A pair of operators will be inserted on a cylinder, separated 2iǫ

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Thermofield double set-up

Consider two non-interacting 2d CFTs, say CFTL and CFTR, with isomorphic Hilbert spaces HL,R Thermofield double state : |Ψβ = 1

• Z(β)
• n

e− β

2 En |nL |nR

|nL is an eigenstate of the hamiltonian HL acting on HL with eigenvalue En (and similarly for |nR). |nL is the CPT conjugate of the state |nR Notation : |nL ⊗ |nR as |nL |nR. Thermal reduced density ρR(β) = trHL (|Ψβ Ψβ|) = 1 Z(β)

• n∈HR

e−βEn |nR n|R ,

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Thermofield double : observables

Single sided correlators are thermal Ψβ| OR(x1, t1) . . . OR(xn, tn) |Ψβ = trHR (ρR(β)OR(x1, t1) . . . OR(xn, tn)) . Two sided correlators : by analytic continuation Ψβ| OL(x1, −t) . . . OR(x′

n, t′ n) |Ψβ =

trHR

• ρR(β)OR(x1, t − iβ/2) . . . OR(x′

n, t′ n)

• .

Will use this observation when computing Renyi entropies

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CFT considerations

As discussed by Morrison & Roberts (see also Hartman & Maldacena) : single sided thermal correlation functions are computed on a single cylinder with periodicity τ ∼ τ + β two-sided correlators involve a path integral over a cylinder with the same periodicity τ ∼ τ + β, where all operators OR are inserted at τ = iβ/2, whereas OL are inserted at τ = 0 Set-up : Consider thermofield double state two finite intervals: A = [y, y + L] in the left CFTL and B = [y, y + L] in the right CFTR perturb the TFD by the insertion of a local primary operator OL acting on CFTL at x = 0 , t− = −tω

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Calculation of SA

SA = − lim

n→1

1 n − 1 log (Tr ρn

A(t))

where Tr ρn

A(t) = ψ(x1, ¯

x1)σ(x2, ¯ x2)˜ σ(x3, ¯ x3)ψ†(x4, ¯ x4)Cn (ψ(x, ¯ x1)ψ†(x4, ¯ x4)C1)n with the insertion points x1 = −iǫ , x2 = y − tω − t−, x3 = y + L − tω − t− , x4 = +iǫ ¯ x1 = +iǫ , ¯ x2 = y + tω + t−, ¯ x3 = y + L + tω + t− , ¯ x4 = −iǫ with conformal dimensions Hψ = nhψ , Hσ = c 24

• n − 1

n

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Conformal maps

1

From the cylinder to the plane ω(x) = e2πx/β

2

Standard map : ω1 → 0, ω2 → z, ω3 → 1 and ω4 → ∞ z(ω) = (ω1 − ω)ω34 ω13(ω − ω4) where the cross-ratio satisfies z = ω12ω34 ω13ω24

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Result

S(n)

A

= c(n + 1) 6 log

• β

πǫUV sinh πL β

• +

1 n − 1 log

• |1 − z|4HσG(z, ¯

z)

• where

G(z, ¯ z) = ψ| σ(z, ¯ z)˜ σ(1, 1) |ψ Using the large c results derived by Fitzpatrick, Kaplan & Walters in the limit n → 1 ∆SA = c 6 log

• z

1 2 (1−αψ)¯

z

1 2 (1−¯

αψ)(1 − zαψ)(1 − ¯

z ¯

αψ)

αψ ¯ αψ(1 − z)(1 − ¯ z)

• where αψ =
• 1 − 24hψ

c

.

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Final result

Analysing the imaginary parts, we reach the conclusions : (z, ¯ z) → (1, 1) for t + tω < y and t + tω > y + L (z, ¯ z) → (e2πi, 1) for y < t + tω < y + L The importance of this monodromy has been emphasized by several groups including Asplund, Bernamonti, Galli & Hartman and Roberts & Stanford ∆SA = 0 , t− + tω < y and t− + tω > y + L ∆SA = c 6 log   β πǫ sin παψ αψ sinh

• π(y+L−t−−tω)

β

• sinh
• π(t−+tω−y)

β

• sinh
• πL

β

• 

 y < t− + tω < y + L

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Calculation of SB

Very similar, but with different insertion points : Tr ρn

A(t) = ψ(x1, ¯

x1)σ(x5, ¯ x5)˜ σ(x6, ¯ x6)ψ†(x4, ¯ x4)Cn (ψ(x, ¯ x1)ψ†(x4, ¯ x4)C1)n with the insertion points x5 = y + L + i β 2 − t+ − tω, x6 = y + i β 2 − t+ − tω ¯ x5 = y + L − i β 2 + t+ + tω, ¯ x6 = y − i β 2 + t+ + tω We always obtain the expected thermal answer at all times SB = c 3 log

• β

πǫUV sinh πL β

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Calculation of SA∪B

Very similar, but with different insertion points : Tr ρn

A∪B(t) = ψ(x1, ¯

x1)σ(x2, ¯ x2)˜ σ(x3, ¯ x3)σ(x5, ¯ x5)˜ σ(x6, ¯ x6)ψ†(x4, ¯ x4)Cn (ψ(x, ¯ x1)ψ†(x4, ¯ x4)C1)n with the insertion points x1 = −iǫ, x2 = y − t− − tω, x3 = y + L − t− − tω, x4 = +iǫ ¯ x1 = +iǫ, ¯ x2 = y + t− + tω, ¯ x3 = y + L + t− + tω, ¯ x4 = −iǫ x5 = y + L + i β 2 − t+ − tω, x6 = y + i β 2 − t+ − tω, ¯ x5 = y + L − i β 2 + t+ + tω, ¯ x6 = y − i β 2 + t+ + tω .

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Strategy

Using conformal maps Tr ρn

A∪B =

• β

πǫUV sinh πL β

• −8Hσ

|1 − z|4Hσ |z56|4Hσ ψ|σ(z, ¯ z)˜ σ(1, 1)σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ where all cross-ratios z, zi are analytically known. ψ|σ(z, ¯ z)˜ σ(1, 1)σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ expected 6-pt function

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S-channel (I)

Let us introduce a resolution of the identity ψ|σ(z, ¯ z)˜ σ(1, 1)σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ =

• α

ψ|σ(z, ¯ z)˜ σ(1, 1) |α α| σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ (z, ¯ z) → (1, 1) for ǫ

β ≪ 1 ⇒ use OPE !!

σ(z, ¯ z)˜ σ(1, 1) ∼ I + corrections in (z − 1)r Or Orthogonality of 2-pt functions ⇒ |α = |ψ dominant Thus, ψ|σ(z, ¯ z)˜ σ(1, 1)σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ ≃ ψ|σ(z, ¯ z)˜ σ(1, 1) |ψ ψ| σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ

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S-channel (II)

Using conformal maps ψ| σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ = |1 − ˜ z5|4Hσ |z56|−4Hσ ψ| σ(˜ z5, ¯ ˜ z5)˜ σ(1, 1)|ψ , we obtain Tr ρn

A∪B ≃

• β

πǫUV sinh πL β

• −8Hσ

|1−z|4Hσ |1 − ˜ z5|4Hσ G(z, ¯ z)G(˜ z5, ¯ ˜ z5)+... Since ˜ z5 = z5, the cross-ratio determining SB, we derive SA∪B = SA + SB, and I(A : B) = 0 This resembles the bulk calculation from two geodesics joining pairs of points in the same boundary !!

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T-channel (I)

We could introduce the resolution of the identity as follows ψ|σ(z, ¯ z)˜ σ(1, 1)σ(z5, ¯ z5)˜ σ(z6, ¯ z6)|ψ =

• α

ψ|σ(z, ¯ z)˜ σ(z6, ¯ z6) |α α| σ(z5, ¯ z5)˜ σ(1, 1)|ψ . (z5, ¯ z5) → (1, 1) for ǫ

β ≪ 1 ⇒ use OPE !!

As before, |α = |ψ dominant contribution !!

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T-channel (II)

In this case, Tr ρn

A∪B ≃

• β

πǫUV sinh πL β

• −8Hσ
• x

1 − x

• 4Hσ

|1 − z5|4Hσ|1 − ˜ z2|4Hσ G(˜ z2, ¯ ˜ z2)G(z5, ¯ z5) + ... where (x, ¯ x) are the cross-ratios computed out of the insertion points of the four twist operators x = z23z56 z25z36 = w23w56 w25w36 = 2 sinh2 πL

β

cosh 2πL

β + cosh 2π(t−−t+) β

= ¯ x ,

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T-channel (III)

For t− + tω > y + L, we derive

SA∪B ≃ c 6 log

• sinh π(t−+tω−y)

β

cosh π(t++tω−y)

β

cosh π∆t

β

• + c

6 log

• sinh π(t−+tω−y−L)

β

cosh π(t++tω−y−L)

β

cosh π∆t

β

• + 2c

3 log

• β

πεUV cosh π∆t β

• + c

3 log β πǫ sin παψ αψ

• where ∆t = t− − t+

To derive this result we used Fitzpatrick, Kaplan & Walters Assumption : conformal block of the identity is dominant

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Mutual information

In the regime t∓ + tω > y + L > y,

I(A : B) ≃ 2c 3 log

• β

πεUV sinh πL β

• − 2c

3 log

• β

πεUV cosh π∆t β

• − c

3 log β πǫ sin παψ αψ

• − c

6 log

• sinh π(t−+tω−y)

β

cosh π(t++tω−y)

β

cosh π∆t

β

• − c

6 log

• sinh π(t−+tω−y−L)

β

cosh π(t++tω−y−L)

β

cosh π∆t

β

• take t− = t+ = 0 and look for t⋆

ω satisfying

I(A : B)(t⋆

ω) = 0

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Scrambling time

Assuming t⋆

ω/β ≫ 1, we obtain

t⋆

ω = y + L

2 − β 2π log β πǫ sin παψ αψ

• + β

π log

• 2 sinh πL

β

• if hψ ≪ c, then

t⋆

ω = y + L

2 + β 2π log πSdensity 4Eψ

• + β

π log

• 2 sinh πL

β

• where we used

β πǫ sin παψ αψ ∼ πEψ Sdensity with Sdensity = πc

3β and Eψ = πhψ ǫ

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Holographic considerations

Main idea & strategy : Static point particle at r = 0 in global AdS3 ds2 = −

• r2 + R2 − µ
• dτ 2 +

R2dr2 r2 + R2 − µ + r2dϕ2 , Holographic entanglement entropy known SA = c 6

• log
• r(1)

∞ · r(2) ∞

R2

• + log 2 cos (|∆τ∞|αµ) − 2 cos (|∆ϕ∞|αµ)

α2

µ

• Map metric to Kruskal coordinates, while boosting the particle, to

describe a free falling particle in eternal BTZ

◮ Use an initial condition ensuring the particle carries the right energy,

from CFT and stress tensor perspective

Map endpoints & compute entanglement entropy

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Holographic comments

Calculations involve many explicit technical details, leading to

1

Exact matching of dominant CFT contributions with the holographic model geodesic calculations

◮ S-channel and T-channel contributions precisely match the two

dominant geodesics computing SA∪B

2

In the limit of large tω :

◮ free falling particle becomes almost null with energy localised at the

horizon

◮ matches the schock-wave descriptions proposed/used by Shenker,

Stanford, Roberts, Susskind, ...

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Relation to quantum chaos & butterfly effect

Kitaev, and later Maldacena, Shenker & Stanford, suggested the use of a large commutator −[W (t), V (0)]2β to diagnose the butterfly effect. Using the regularised quantity, A = −tr

• y2 [W (t), V (0)] y2 [W (t), V (0)]
• where

y4 = e−β H Z(β) it follows A = tr

• y2 W (t)V (0) y2V (0)W (t)
• + tr
• y2 V (0)W (t) y2W (t)V (0)
• −

F(t + i β 4 ) − F(t − i β 4 ) F(t) = tr (yV (0)yW (t)yV (0)yW (t)) . Notice F(t) involves out of time ordered (OTO) correlators.

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Relation to quantum chaos & butterfly effect

Using the thermo-field double state : First two terms are order 1, for all t, since they are norms of states Growth of the regularised commutator, requires F(t ± iβ/4) to become small F(t) = Ψ|VLVR|Ψ with |Ψ =

1

√

Z(β)

• m,n e−β(Em+En)/4W (t)nm|mL|nR

small t, F(t) is order one due to its nearly maximally entangled nature as t increases, correlations can get destroyed and F(t) decreases ⇒ quantum chaos ∼ destruction of these correlations.

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Holographic considerations(IMPROVE)

Time ordered correlators : V (0)V (0)W (t)W (t) ∼ VV WW + O

• e−t/td
• where td ∼ β is controlled by BH quasi-normal modes.

Holographic calculations determine OTOs to behave like (Shenker, Stanford, Roberts, Susskind) W (t)V (0)W (t)V (0)β = f0 − f1 N2 e2πt/β + O(N−4) OTO decays at time scales t⋆ ∼ β

2π log N2.

Characteristic feature for holographic theories. Question : Any CFT evidence for this behaviour ?

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Large c 2d CFT considerations

Euclidean correlators V †(z1, ¯ z1)V (z2, ¯ z2)W †(z3, ¯ z3)W (z4, ¯ z4) V †(z1, ¯ z1)V (z2, ¯ z2)W †(z3, ¯ z3)W (z4, ¯ z4) = A(z, ¯ z) have branch cuts from z ∈ (1, ∞) & OTOs defined by analytic continuation z1 = e

2π β iε1 ,

¯ z1 = e− 2π

β iε1

z2 = e

2π β iε2 ,

¯ z2 = e− 2π

β iε2

z3 = e

2π β (t+iε3−x) ,

¯ z3 = e

2π β (−t−iε3−x)

z4 = e

2π β (t+iε4−x) ,

¯ z4 = e

2π β (−t−iε4−x)

Chaos explored in the regime t − x ≫ β z = z12z34 z13z24 ≈ −e− 2π

β (t−x) ε⋆

12ε34 → 0 ,

¯ z z fixed

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Large c 2d CFT considerations

Assuming large c, sparse spectrum of light operators absence of light single-trace operators of spin s > 2 Perlmutter, building on Roberts & Stanford VW (t)VW (t)β VV βW (t)W (t)β ∼ 1 − i c ε⋆

12ε34

e− 2π(t−x)

β

f (e− 4π

β x) + . . .

Scrambling time ∼ scale at which expansion breaks down t⋆ ∼ β 2π log c

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Butterfly effect ⇒ scrambling (in QM)

Hosur, Qi, Roberts & Yoshida showed that averaging OTOs over a complete basis of operators in the subsystems A and D |OD(t) OA(0) OD(t) OA(0)| ∼ 2−S(2)

A∪C

Since SA∪C ≥ S(2)

A∪C

I(A : C) ≤ 2a − log2 ǫ with ǫ ∼ f0 − f1 N2 eλL t where we already assumed the system is quantum chaotic. Introducing N2 ∼ eλL t⋆, then we find I(A : C) ≤ 2a − ♮ eλL (t−t⋆) + . . . If system is chaotic, QI magnitudes relevant to scrambling approach their Haar-scrambled values.

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Scrambling ⇒ quantum chaos ?

Question : what about integrable 2d CFTs ? Intuition : OTO should remain of order one (lack of quantum chaos) Caputa, Numawara & Veliz-Osorio (see also Qi & Gu) find this behaviour for 2d rational CFTs ∃ scrambling in SU(N)k WZW models in the large c limit

• rder of limits concerning c → ∞ ??

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